I'm learning linear regression using "A Primer on Linear Models" by John F. Monahan. On page 14, it says
$$Q(b) = (y-Xb)^T(y-Xb)=y^Ty − 2y^T Xb + b^T X^T Xb$$ Where $y$ is a $N ×1$ vector of observed responses, $X$ is a $N × p$ matrix of fixed constants, $b$ is a $p × 1$ vector of fixed but unknown parameters.
by developing brackets I can get $-y^TXb-(Xb)^Ty$. But I don't know from this how to get $− 2y^T Xb$ term. Why is it possible to just flip $(Xb)^Ty$ to $y^TXb$?
Also, on the same page 14,
$$\frac{d Q}{d b}=-2X^Ty + 2X^TXb = -2X^T(y-Xb)$$
I don't understand how I can get this result.
It is well known that:
$$(MN)^T = N^T M^T.$$
Starting from this point, let's consider the transpose of $y^T Xb:$
$$(y^T Xb)^T = (y^T (Xb))^T = (Xb)^T (y^T)^T = (Xb)^T y.$$
Then,
$$-y^TXb-(Xb)^Ty = -2y^TXb = -2(Xb)^Ty.$$